Minimax optimal submatrix detection: Sharp non-asymptotic rates

2026-05-10

AI Summary

This study addresses the problem of detecting a mean-shifted submatrix of unknown location and size $s_1 \times s_2$ embedded in a high-dimensional Gaussian matrix. By constructing a novel test statistic, the authors establish matching minimax lower and upper bounds for the minimal detectable signal strength $\mu^*$ under a non-asymptotic framework, valid for arbitrary ambient dimensions and sparsity configurations. Furthermore, they propose an adaptive detection procedure that requires no prior knowledge of the sparsity level, thereby overcoming restrictive assumptions on parameter scaling present in existing methods. The proposed approach achieves minimax optimality across a broad range of settings while effectively controlling both Type I and Type II error probabilities.

Abstract

We consider the problem of detecting a hidden submatrix of size $s_1 \times s_2$ in a high-dimensional Gaussian matrix of size $d_1 \times d_2$. Under the null hypothesis, the observed matrix has i.i.d.\ entries with distribution $N(0,1)$. Under the alternative hypothesis, there exists an unknown submatrix of size $s_1 \times s_2$ with i.i.d.\ entries with distribution $N(μ, 1)$ for some $μ>0$, while all other entries outside the submatrix are i.i.d.\ $N(0,1)$. Specifically, we provide non-asymptotic upper and lower bounds on the smallest signal strength $μ^*$ that is both necessary and sufficient to ensure the existence of a test with small enough Type I and Type II errors. We also derive novel minimax-optimal tests achieving these fundamental limits, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. Our proposed detection procedure is a careful combination of novel test statistics which may be of independent interest. In contrast with previous work, which required restrictive assumptions on $d_1, d_2, s_1$ and $s_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters.

Problem

• submatrix detection
• minimax optimality
• non-asymptotic rates
• Gaussian matrix
• signal detection

Innovation

• minimax optimality
• submatrix detection
• non-asymptotic analysis
• adaptive testing
• Gaussian matrix